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I’ve always aimed to make Kangaroo a specifically architectural physics engine. While it shares many characteristics with similar engines used for other purposes, such as games and animation, it has some features that are uniquely suited to designing buildings.

Form-finding and physics-based-modelling often result in curved shapes, with an elegant and natural appearance which is something I greatly enjoy about these methods.

However, the fact remains that most existing architectural geometry consists of straight lines and right-angles.

So the latest addition I have made to Kangaroo is for interaction between boxes, which remain aligned with the global XYZ directions. As shown in the video above, this can be combined with other forces such as gravity and attraction, causing the boxes to cluster together, and they will always remain orthogonal.

Because of the constraint to the component directions, this actually removes the need for calculating any square roots for distances, which is usually a significant part of the physics computation, so it is simple and fast.

I’ve also added an option to allow the boxes to alter their proportions in order to maintain a fixed volume. So as they get pushed on by neighbouring boxes or other forces they may get squashed or elongated, but they keep their contained volume the same (Movement could also be constrained in the vertical direction, so areas are fixed).

Some people have already used Kangaroo for arranging rooms (for example see Marc Syp’s project Realtime Physics for Space Planning), using systems of weighted springs and colliding spheres to balance complex networks of  adjacency requirements between different functional areas of buildings. I’m hoping this new functionality could be useful in taking these ideas further…

All of these triangles are identical and equilateral:

In architectural geometry over the last few decades, a common topic of research has been how to build and clad doubly curved surfaces in an efficient way.

While computer aided manufacturing has made it possible to make buildings where every panel has slightly different dimensions, there are still often significant benefits in fabrication and assembly if identical panels can be used.

However, it is a geometric impossibility to smoothly cover a doubly curved surface with many perfectly identical panels, all meeting exactly edge-to-edge. The typical way of dealing with this is to have many variations in the shapes of the panels. Non-uniformity in the connectivity of the mesh (ie. some irregular vertices) can help reduce the magnitude of these variations (see my previous post for more on this), but still leaves many different panel shapes.

An approach I’ve been exploring a bit recently is to allow some gaps to open up between panels in a controlled way, but still keeping vertices connected. By each panel rotating slightly in a particular alternating clockwise/counter-clockwise pattern they are able to expand and contract to allow for the curvature of the surface. To keep a high ratio of panel to gap area, this expansion must be limited, and well positioned irregular vertices (the 4 pointed stars in the image above) can help with this.

This can be done with various tiling types – squares, triangles, and hexagon/triangle combinations. However, to allow this alternating twist pattern, it must be one which can be coloured in a checkerboard fashion, which also means every internal vertex must be surrounded by an even number of faces.

Of course, if we want to make such a structure into a closed skin, then something needs to fill in the gaps, which do all vary in shape, but if these are made from a material which is easier to cut, I think there could still be a significant overall benefit. For example, mass-produced identical triangular or square glass panels, with varying aluminium ‘stars’ or ‘diamonds’ for the gaps.

Designing such panelizations is a good application of a computational physics design tool like Kangaroo. Indeed I was inspired by the sculptures of Ron Resch, who produced some similar forms as an emergent result of physical processes of folding paper.

Ron Resch in his studio. Photo from Joan Michaels Paque via Eric Gjerde

(Eric also pointed out to me this wonderful video of some more of Resch’s work)

See also the ‘Tile Magic’ series here for some other possible tilings that could be used

A more recent inspiration is Haresh Lalvani who has produced some related forms, by cutting thin slits in sheet steel, allowing it to twist and expand in a similar way when deformed.

Sculpture by Haresh Lalvani

It is not always easy to predict what surfaces it is possible to panel in such a way, and only by setting up the physical constraints and manipulating the surface do we see how it can deform and get a feel for its possibilities.

At a small scale we can actually do this with the real material, but at a larger scale, if we wanted to pre-calculate the shape and the dimensions of the varying gap pieces beforehand then the digital-physical form-finding is vital.

Amazingly Ron Resch actually even wrote some computer programs back in the 60s to do some fairly similar iterative form-finding and even produce 3d animations. (See in the video linked to earlier from about 28:00 to 38:00)

Such examples of what was possible even with relatively primitive technology are an inspiration to push things further today…

 

 

 

As regular readers of this blog will know, I’m passionate about the use of relaxation and force-based methods for optimizing geometry in a very interactive way.

There is a great variety of form-finding that can be done by assigning physical forces as interactions between sets of particles. However, in my investigations so far, the topology of these interactions has usually remained fixed. So the overall shape changes dramatically, but the number of points -and the underlying network of which point interacts with which- remains the same throughout the simulation.

I have written about topology optimization before, in my post on self organizing structures, but the videos above use a fundamentally different approach than any of the examples shown there, as this new work is based on purely local interactions, which reconfigure their connectivity as the geometry changes.

I did experiment a while back with a different technique for dynamic remeshing based on repelling/colliding particles or spheres (see this video, skip to around 1:27 for the mesh generation), but this was limited by a number of factors – although the spheres only exert force on each other when they are closer than a certain distance, they still need to be constantly checked against all the other spheres to see if they are within this distance. Checking every sphere against every other one means the number of interactions increases as the square of the number of particles, which becomes slow for even modest meshes of a few thousand vertices.

There are various ways of significantly reducing this complexity and cutting down the number of distance calculations to be performed at each iteration (such as partitioning the space into cubic cells, and only checking for interactions between particles in the same or neighbouring cells).  However, there still remains the question of how to remove or add spheres if they get too close or too spread out. Then even when a fairly equal distribution of points across the surface has been achieved, turning this into a clean mesh can still be tricky.

Instead, using the mesh connectivity itself from the very beginning as the network of which vertices interact with which, and updating it iteratively and locally based on its changing geometry, solves all of these problems at once. This connectivity update can be done by using repetitions and combinations of just 3 essential moves :

edge flip

edge collapse

edge split

(or alternatively, one can use edge flip, vertex insertion and vertex removal)

vertex insertion/removal

I’m always using a triangulated mesh here. In the 2nd and 3rd videos at the start, the underlying physics is still working with triangles, but I’ve shown the dual of this, which exchanges the roles of faces and vertices, turning a mesh of mostly triangles into one of mostly hexagons:

Notice that the irregular vertices of the triangle mesh (ones which have 5 or 7 connected edges instead of the regular 6) correspond to pentagons and heptagons in the dual. Finding the right number and placement of these irregular vertices is an important part of making a good mesh. There is a close relationship between these irregular vertices and the overall topology and curvature of the mesh. Euler’s polyhedron theorem gives us a precise relationship between the number of faces/edges/vertices, and the genus of the mesh (F-E+V=2-2g). This is linked to Descarte’s polyhedron theorem, which relates the total angle defect at the vertices to the genus, and is itself a discrete version of the continuous Gauss-Bonnet theorem which describes a similar relationship for the integral of Gaussian curvature in the smooth case.

Keeping a good quality triangular mesh (close to even sized equilateral elements, with no obtuse, skinny or degenerate ones) can be very useful for many other types of optimization and simulation, as well as an advantage for fabrication if it is to be built as a physical structure.

I’m thinking I could also vary the remeshing rules depending on whether the priority is regularity of geometry, or regularity of connectivity. For example, on a sphere, a geodesic dome has a small number of irregular vertices (this is sometimes referred to as a semi-regular mesh), but a fair amount of variation in edge length, whereas solutions to the Thomson or Tammes problems reduce the variation of distance, but have larger numbers of irregular vertices.

If the triangles all have identical edge lengths, then all of the angle defect is concentrated at the irregular vertices, whereas if the edges can vary slightly, the angle defect can be spread out across the mesh. As mentioned above, by Descarte’s theorem, the total angle defect is fixed, but if the number of vertices is increased it can be divided between more of them. Subdividing each triangle (using a smoothing scheme such as Loop subdivision) does not alter the configuration of irregular vertices, but reduces the angle defect at each vertex (and in the smooth limit it approaches zero as the number of vertices approaches infinity).

The remeshing can also be based on other criteria than just equal triangles. Reducing triangle size in high curvature areas is an obvious one, but I have a hunch there could also be some interesting ways of basing it not just on geometry, but on stresses, and using it for structural optimization.

I find remeshing fascinating because the same geometric rules and operations have relevance across so many different disciplines and at a variety of scales.

For example, in the carbon nanostructures graphene and nanotubes, which are hexagonal arrangements of atoms, there is a much studied crystallographic defect which occurs called the Stone-Wales defect – where instead of all hexagons we get 2 pentagons and 2 heptagons. It has important implications for the material’s mechanical and electrical properties. If we look again at our edge-flip move on a regular triangular mesh, and its effect on the dual, we see that it is exactly the Stone Wales defect!

There are even hypothetical carbon nanostructures (proposed by Mackay and Terrones)  in the shape of doubly curved triply periodic minimal surfaces known as Schwarzites.

(from the paper Curved nanostructured materials)

As the bond lengths between the Carbon atoms are very rigid, the only way these curved structures can be formed is through the variations in mesh connectivity.

Because such complex geometries are only dependent on the way in which many identical elements are connected, rather than variation in the elements themselves, they can be modelled using simple materials (no laser cutting involved – just clever assembly!):

(by Bih-Yaw Jin from the beaded molecules)

(by Dimitry Tishchenko)

(by Loop.ph)

Going down to even smaller scales, we can even find the use of similar ideas about mesh connectivity in theories of loop quantum gravity, spin foams, and Regge calculus (for example, see Canonical Simplicial Gravity by Dittrich and Höhn, or The Feynman diagramatics for the spin foam models). Pachner moves or bistellar flips generalize the mesh moves described earlier to higher dimensional simplices.

        

(images from Canonical Simplicial Gravity)

So as well being potentially useful for design, meshes and their properties have profound relevance to our understanding of the nature of space and curvature, because of the way they link the discrete and the continuous.

I’ll be continuing to work on these remeshing tools and their combination with other types of relaxation and optimization. These recent developments are not publicly available for now, but I’ll update here with any news.

Here are a few more references on remeshing:

Dynamic Remeshing and Applications

A Remeshing Approach to Multiresolution Modeling

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also – a general update :  Around a year ago I started working full time for Foster+Partners in the Specialist Modelling Group. This has been a really great experience, working on (and applying Kangaroo on) some big and exciting projects, but core Kangaroo development (and posts here) did necessarily slow down a bit during this time. However, I have recently left F+P and things will be changing ! More news to follow…

      

      

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